= Solution
Use the <Boussinesq approximation> and a <top-hat plume model>. Let $v(z)<0$ be the uniform ambient return <velocity> outside the plume. Negligible source <volume flux> and an impermeable cylindrical wall require zero net transport through every horizontal section:
$$
\boxed{b^2w+(R^2-b^2)v=0,\qquad v=-\frac{b^2w}{R^2-b^2}.}
$$
The entrainment speed must be specified relative to that moving ambient. Taking it to be $\alpha(w-v)$ gives
$$
\boxed{\frac{d}{dz}(b^2w)=2\alpha b(w-v).}
$$
The surrounding unmodified fluid has zero <buoyancy> relative to the initial fluid. Neglecting thermal or compositional loss therefore gives
$$
\boxed{\frac{d}{dz}(b^2wg')=0.}
$$
Let $B_s$ denote the conserved specific source <buoyancy flux>: $B_s=B_o/\pi$ if $B_o$ is the actual integrated source flux, or $B_s=B_o$ if the source strength has already been normalized by $\pi$.
To obtain the usual integral-model singularity one must also state a <pressure> closure. Here take the whole-section mean <pressure> to have only its reference ambient <hydrostatic pressure>, and neglect wall force and extra vertical turbulent-stress fluxes in the integral budget. Internal <momentum> transfers between plume and ambient cancel in that whole-section balance. The remaining upward force is the plume <buoyancy>, giving
$$
\boxed{\frac{d}{dz}\left[b^2w^2+(R^2-b^2)v^2\right]=b^2g'.}
$$
The two <momentum fluxes> add, even though $v$ is negative: vertical <momentum> transported downward also contributes $v^2$ to the flux through an upward-oriented horizontal section. These three flux balances, with the algebraic return-velocity constraint, govern $b,w,g'$ in the <confined plume with compensating return flow>.
The pressure-neglecting closure is an integral approximation, not a claim that the return fluid has zero acceleration. If a mean <pressure> departure $p_*(z)$ is retained, the whole-section <momentum> balance instead contains $-(R^2/\rho_0)p_*'(z)$, along with any wall-force or stress contribution. The entrainment hypothesis alone does not specify that function. The singularity calculated below belongs to the stated pressure-neglecting model; it is not forced by continuity alone or asserted to be an infinite <velocity> in the real fluid.
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